7 THE STURM-LIOUVILLE PROBLEM 357

(c) Under the assumptions of (a), let zl,..., zn _ i be n — 1 arbitrary twice continuously
diflferentiable functions in I, and let B(zl5..., zn~i) be the intersection of A and of
the n — I hyperplanes O| z*) = 0 (1 < k < n — 1). Show that in B(ZI, ..., zn_i), the
function 3> reaches a minimum p(z^ ..., zn_i) at a point of fife,..., zn_i), and that
Art is the l.u.b. of p(zlt..., zn,i) when the zt vary over the set of twice continuously
differentiable functions in I (the "maximmimal" principle; same method as in (a) to
prove the existence of the minimum; the inequality is proved by the same method as
in Section 11.5, Problem 8). Extend the result to the cases ktk2 = 0.
3. (a) One considers in the same interval I two linear differential equations of the
second order y" — qty + Xy = 0, y" — q2y 4- Xy — 0, with the same boundary con-
ditions (11.7.2); let (X(nl)), (A<2)) be the two strictly increasing sequences of eigenvalues
of these two Sturm-Liouville problems. Show that if q± ^ qi, then AJ,1' ^ A(n2) for every
n, and if \qi(t) — ^(0| < M in I, then \X^ — Ai2)| ^ M for every n (use the maximinimal
principle),
(b) Conclude from (a) that there is a constant c such that

for every n, with /= b — a. (Study the Sturm-Liouville problem for the particular case
in which q is a constant.)

4. (a) Let y be any solution of (1 1 .7.3) in I = [a, b] for A > 0. Show that there are two
constants A, to such that y is a solution of the integral equation

(*) XO == A sin ^X(t + o>) -f -p rq(s)y(s) sin ^X(t - s) ds.

>Aja

Show that there exists a constant B independent of A, such that A2 ^ B0> | y) (use
Cauchy-Schwarz in order to majorize the integral on the right-hand side of (*)).
(b) Deduce from (a) that if, in the Sturm-Liouville problem, kvki =£ 0 or ki = k2 = 0,
then there are two constants C0 , Q, such that, for every 72, and every /el

\q>n(t) - ^2/l sin Xnt\ ^ C0/n
and

» cos V^^l < GI with l—b-

(use (a), and the result of Problem 3(b)). What is the corresponding result when only
one of the constants kl9 k2 is 0?r £.)